Nonlocal degenerate parabolic hyperbolic equations on bounded domains

TL;DR

This work develops an entropy framework for degenerate mixed-type nonlocal parabolic-hyperbolic equations on bounded domains, incorporating Dirichlet exterior data and inflow boundary effects. It introduces a domain-adapted entropy formulation with a nonlocal boundary condition, plus a decomposition $\mathcal{L}=\mathcal{L}^{<r}+\mathcal{L}^{\geq r}$, and proves a sharp $L^1$-contraction type uniqueness result for bounded entropy solutions via a careful doubling of variables and boundary-trace analysis. Existence is established in a companion paper, and key a priori results include $L^\infty$ bounds, finite energy, strong initial traces, and a nonlocal boundary trace, enabling a robust well-posedness theory on bounded domains. Overall, the paper extends nonlocal entropy theory from the whole space to bounded domains and links nonlocal diffusion with hyperbolic inflow/boundary behavior, highlighting how energy estimates emerge from the entropy formulation rather than being assumed apriori.

Abstract

We study well-posedness of degenerate mixed-type parabolic-hyperbolic equations $$ \partial_tu+\text{div}\big(f(u)\big)=\mathcal{L}[b(u)] $$ on bounded domains with general Dirichlet boundary/exterior conditions. The nonlocal diffusion operator $\mathcal{L}$ can be any symmetric L{é}vy operator (e.g. fractional Laplacians) and $b$ is nondecreasing and allowed to have degenerate regions ($b'=0$). We propose an entropy solution formulation for the problem and show uniqueness of bounded entropy solutions under general assumptions. Existence of solutions is shown in a separate paper. The uniqueness proof is based on the Kružkov doubling of variables technique and incorporates several a priori results derived from our entropy formulation: an $L^\infty$-bound, an energy estimate, strong initial trace, weak boundary traces, and a \textit{nonlocal} boundary condition. Our work can be seen as both extending nonlocal theories from the whole space to domains and local theories on domains to the nonlocal case. Unlike local theories our formulation does not assume energy estimates. They are now a consequence of the formulation, but as opposed to previous nonlocal theories, they play an essential role in our arguments.

Theorems & Definitions (59)

• Remark 2.1
• Definition 2.2: Entropy solution
• Remark 2.3
• Lemma 2.4: $L^\infty$-bound
• Lemma 2.5: Time continuity at $t=0$
• Proposition \ref{prop: finiteEnergy}: Energy estimate
• Remark 2.6
• Proposition \ref{prop: boundaryIntegrability}: Boundary integrability
• Proposition \ref{prop: weakTrace}: Weak trace
• Remark 2.7